14 variable sampling plan
TRANSCRIPT
Sampling Techniques by
Variables
Topic Outcome: At the end of this topic, students will be able to:
Discuss the advantages and disadvantages of variable sampling.
Apply Process Fraction Nonconforming Type of variable sampling.
Use ANSI/ASQ Z1.9 – 1993.Describe the Shainin Lot Plot Plan.
Outline: An Introduction. Types of Sampling Plan.
- Shainin Lot Plot Plan.
- ANSI/ASQ Z1.9
Variability UnknownStandard Deviation MethodRange Method
Variability known
An Introduction When a quality characteristic is measurable on a
continuous scale, and is know to have a distribution of a specific type (e.g. normal distribution); it is possible to use a sampling based on sample measurements such as:
Mean and
Standard Deviation.
Variables Sampling Plans.
Population
Sample
LSL USL
LSL USL
Compare
Compare to required Quality LevelACCEPT REJECT
Percent nonconforming below and above specifications.
LSL USL
pu
+-pL
Advantages of Variable Sampling Plan:
Same protection with smaller sample size.
Feedback of data on Process.
Extent of conformity given weightage.
Errors in measurement more likely detected.
Disadvantages of Variable Sampling Plan:
Applicable to only one quality characteristic at a time.
Higher inspection cost – Measurement.
Higher clerical cost – Calculation.
Possibility of no nonconformity in a rejected lot.
Dependence on assumption of distribution shape.
Types of Variable Sampling PlanVariable Sampling Plan
Percent nonconforming Process Parameter
To determine the proportion of product that
is outside specification
To control the average and standard deviation of
the distribution of the product to specified level
E.g.:Shainin lot plotANSI/ASQ Z1.9-1993
E.g.:Shainin lot plotANSI/ASQ Z1.9-1993
E.g.:Acceptance control chartSequential sampling for variablesHypothesis testing
Shainin Lot Plot Plan Developed by Dorian Shainin (Hamilton Standard Division
of United Aircraft Corporation, 1950).
The plan uses a plotted frequency distribution (histogram) to evaluate a sample for decisions (accept/reject a lot).
It is a practical plan for in-house inspection & receiving inspection.
Advantages:
Applicable to both normal and non-normal frequency distributions.
Simple to use.
The method for obtaining the lot plots is as follows:
A random sample of 10 subgroups (g) of 5 each for a total of 50 items is obtained from the lot.
The average, X-bar and range, R, are calculated for each subgroup.
A histogram is constructed (number of cell: 7-16).
Calculate the average of averages, X-double bar, and average of ranges, R-bar.
Calculate the upper and lower lot limits (ULL & LLL)
Lot Plotting Method
Q&A
9.957.973
5.99326.2
)37.1)(3(7.97
3
326.2)37.1)(3(
2
2
d
RXLLL
d
RXULL
37.110
7.13
7.9710
8.976
g
RR
g
XX
Chart for Ranges:Factor for Central Line d2
3 control limits
Fig 10-14: Lot plot histogram
Decision is based on a comparison of the lot plot with 11 different types of lot plots.
Lot Plot Evaluation
Type Distribution Comments
1
normal
-within spec.limits
- accepted without calculate lot limits.
2-lot limits within spec. limits.
-accept.
3 -lot limits outside spec. limits.
-percentage of product beyond spec is obtained.
-Review board determines the final disposition of the stock.4
5nonnormal
skrewed
6 Lot was screened & sorted
7 Bimodal condition
8nonnormal Lot was screened & sorted
9
10 Bimodal condition
11 Stray values
Once learned, the lot plot procedure is relatively simple and has resulted in improved quality and lower inspection cost.
Unacceptable lots are returned to the producer, and this action will cause a subsequent improvement in quality.
Inspectors can accept lots; however, disposition of unsatisfactory lots is left to a material-review board.
Many users of the lot plot method have modified the Shainin method for their own situation.
The major criticism of the plan is that the shape of the plot does not always give an accurate indication of the true distribution. Shainin states that the plot is close enough to have no practical effect on the final decision, or if there are any errors, they are in a safe direction.
Summary
ANSI/ASQ Z1.9 MIL-STD-414 ASQ closely match ANSI/ASQ Z1.4 and
ISO/DIS 3951. Indexed by AQL (0.10-10.0%). Assumption: normally distributed random variable. Composition of the standard: 9 different procedures to
evaluate a lot.Variability Unknown
(Range Method)Variability Unknown
(Standard Deviation Method)Variability known
Single Specification Double Specification
Form 1(k method)
Form 2(m method)
Form 2(m method)
LSL orUSL
•LSL andUSL•1AQL or2AQL
Quality Index k Quality Index p0 M
k = Acceptability Constant; M = Maximum Allowable Percent Nonconformity
It is divided into 4 sections: Section A: General description of sampling plan, sample
size, code determination, OC curves.
Section B: Unknown Variability – Standard Deviation
Method.
Section C: Unknown Variability – Range Method.
Section D: Know Variability.
It could apply either with a single specification limit or two specification limits.
Specification limits: It is the requirement that a quality characteristic should meet.
This requirement may be expressed as an upper spec. limit
(USL); or a lower spec. limit (LSL), called herein a single
specification limit; or both upper and lower specification limits,
called herein a double specification limit.
For single specification Form 1 and Form 2 give identical end results.
Form 1: Decision on Acceptability is to compare calculated Quality Index to Acceptability Constant, k.
Form 2: Decision on Acceptability is to convert calculated Quality index to Percent Nonconformity in lot p, and then compared to Maximum Allowable Percent Nonconformity, M.
IMPORTANT (Normality Assumption): This standard assumes the underlying distribution of
individual measurements to be normal in shape. Failure of this assumption affect OC curves and probabilities based on these curves. affect the estimate of percent nonconforming calculated from mean and standard deviation. The assumption should be verified prior to use of the standard. A variety of statistical tests and graphical techniques are available.
Relative samples are designated by code letters. The sample sizes code letter depends on the inspection
level and lot size. There are 5 inspection levels: General Levels I, II, and III Special Levels: S3 and S4 The sample size code letter applicable to specified
inspection level and for lots of given size shall be obtained from Table A-2
Determination of Sample Size
Note: Unless otherwise specified, inspection Level II shall
be used. Inspection Level I may be specified when less
discrimination is needed. Inspection Level III may be specified for greater
discrimination. Level S3 and S4 may be used when relatively small
sample sizes are necessary and large sampling risks can be tolerated.
Switching Rules for ANSI/ASQ Z1.9
• Preceding 10 lots Accepted with total nonconforming less than limit number,
• Steady production, and• Approval from
responsible authority.
• Lot not accepted, or• Lot accepted but
nonconformities found lie between Ac and Re of the plan, or
• Irregular production, or• Other conditions
warrant.
• 2 out of 5 consecutive lots not accepted
• 5 consecutive lots accepted
• 10 consecutive lots Remain on Tightened
• Discontinue inspection
NORMAL TIGHTENEDREDUCED
START
There are two parts in this method Single Specification Method Form 1 and 2. Double Specification Method Form 2. Single Spec. Method – Form 1
A) Determine sample size code (Table A-2) by using lot size and inspection level.
B) Select plan from Master Table B-1 and B-2. Obtain sample size, n, and the acceptability constant, k.
C) Select at random the sample of n units from the lot; inspect and record the measurement of each unit.
D) Compute x-bar,sample standard deviation, s, upper/lower specification limit.
E) If the upper/lower spec limit k lot accepted.
Variability Unknown – Standard Deviation Method
Single Spec. Method – Form 2 It converts the computed results into percent
nonconforming (p0) into lot through Table B-5 and compares it with Maximum Allowable Percent Nonconforming, M.
A) Determine sample size code (Table A-2) by using lot size and inspection level.
B) Select plan from Master Table B-3 and B-4. Obtain sample size, n, and M.
C) Select at random the sample of n units from the lot; inspect and record the measurement of each unit.
D) Compute x-bar, sample standard deviation, s, Quality Index, QU or QL.
E) If lot percent nonconforming, pu or pL M lot accepted.
Variability Unknown – Standard Deviation Method
Table B-5
The minimum temperature of operation for a certain device is specified as
180oC. A lot of 40 items is submitted for inspection where inspection level
II, normal inspection,and AQL = 1.0% are the criteria. (Form 1 & 2)
From Table A-2, the code letter D, which gives a sample size n=5
(Table B-1). The temperatures for 5 samples are 197, 188, 184, 205, and
201oC.
Q & A
53.1,tan
70.180.8
180195,
80.815
125.190435.190
1
1955
201205184188197
22
ktConsityAcceptabils
LXQindexqualitylow
nnX
Xs
Cn
XX
L
o
QL k Accepted
Estimated a lot percent nonconforming below L: pL
From Table B-5, pL = 0.66%
Maximum allowable percent nonconforming, M
From Table B-3, M=3.33%
pLM accepted
The maximum temperature of operation for a certain device is specified
as 208oC. A lot of 40 items is submitted for inspection where inspection
level II, normal inspection,and AQL = 1.5% are the criteria. (Form 1 & 2)
From Table A-2, the code letter D, which gives a sample size n=5
(Table B-1).
Q & A
40.1,tan
48.180.8
195208,
80.815
125.190435.190
1
195
22
ktConsityAcceptabils
XUQindexqualitylow
nnX
Xs
Cn
XX
U
o
QU k Accepted
Estimated a lot percent nonconforming above U: pU
From Table B-5, pU = 4.22%
Maximum allowable percent nonconforming, M
From Table B-3, M=5.83%
pUM accepted
Double Spec. Method – Form 2 It can be either:
One AQL value for both Upper and Lower Specification Limit Combined, or
Different AQL values for Upper and Lower Specification Limit.
Variability Unknown – Standard Deviation Method
Double Spec. Method – Form 2 One AQL value for both Upper and Lower Specification
Limit Combined
Variability Unknown – Standard Deviation Method
A) Determine sample size code (Table A-2) by using lot size and inspection level.
B) Select plan from Master Table B-3 and B-4. Obtain sample size, n, and M
C) Select at random the sample of n units from the lot; inspect and record the measurement of each unit.
D) Compute x-bar, sample standard deviation, s, Quality Index, QU or QL.
Determine the estimated lot percent nonconforming, p=pL+pU (Table B-5).
E) p M lot accepted.
The minimum and maximum temperatures of operation for a certain device is specified as 180oC and 209oC. A lot of 40 items is submitted for inspection where inspection level II, normal inspection,and AQL = 1.0% are the criteria.
From Table A-2, the code letter D, which gives a sample size n=5 (Table B-1). The temperatures for 5 samples are 197, 188, 184, 205, and 201oC.
Q & A
Estimated a lot percent nonconforming below L: pL
From Table B-5, pL = 0.66%
70.180.8
180195,
80.815
125.190435.190
1
1955
201205184188197
22
s
LXQindexqualitylow
nnX
Xs
Cn
XX
L
o
)60.1(59.180.8
195209say
s
XUQU
Estimated a lot percent nonconforming above U= pU
From Table B-5, pU = 2.03%
The lot meets acceptance criteria if pL+pU M
From Table B-3, M=3.33%
(0.66+2.03)% 3.32 accepted
Double Specification Method Form 2. Different AQL values for Upper and Lower Specification
Limit. A) Determine sample size code (Table A-2) by using
lot size and inspection level. B) Select plan from Master Table B-3 and B-4. Obtain
sample size, n, Mu (AQL for upper spec limit) and ML(AQL for lower spec limit).
C) Select at random the sample of n units from the lot; inspect and record the measurement of each unit.
D) Compute x-bar, sample standard deviation, s, Quality Index, QU or QL.
Variability Unknown – Standard Deviation Method
E) Determine the estimated lot percent nonconforming, pL and pU (Table B-5), and p (= pL + pU )
F) Accept the lot if the following 3 conditions are met:pU MU
pL ML
p (MU or ML, which ever is larger)
The minimum and maximum temperatures of operation for
a certain device is specified as 180 and 209oC. A lot of 40
items is submitted for inspection where inspection level II,
normal inspection,and AQL = 1.0% for upper and
AQL=2.5% for lower specification limits are the criteria.
From Table A-2, the code letter D, which gives a sample
size n=5 (Table B-1). The temperatures for 5 samples are
197, 188, 184, 205, and 201oC.
Q & A
Line Information needed Value Obtained Explanation
1 Sample size, n 5
2 Estimated Lot Standard Deviation, s 8.81
3 Sample mean, X-bar 195
4 Upper Specification Limit, U 209
5 Lower Specification Limit, L 180
6 Quality Index: QU 1.59
7 Quality Index: QL 1.70
8 Est. Lot Percent Ncf above U, pU 2.19% Table B-5
9 Est. Lot Percent Ncf below L, pL 0.66% Table B-5
10 Total Est PercentNcf, p 2.85%
11 Max. Allowable Percent Ncf above U, MU 3.32% Table B-3
12 Max. Allowable Percent Ncf below L, ML 9.80% Table B-3
13 Acceptability Criteria:
a) Compare pu with MU
b) Compare pL with ML
c) Compare p with ML
2.19%<3.32%
0.66%<9.80%
2.85%<9.80%
ANSI/ASQ Z1.9
Variability Unknown(Range Method)
Variability Unknown(Standard Deviation Method)
Variability known
Single Specification Double Specification
Form 1 Form 2 Form 2
LSL orUSL
•LSL andUSL•1AQL or2AQL
Quality Index k Quality Index p0 M
k = Acceptability Constant; M = Maximum Allowable Percent Nonconformity
Basic approach is similar to those of Standard Deviation Method.
Master Tables used are Tables C-1 to C-5, instead of B-1 to B-5.
Quality indices for Form 1 are calculated as follows:
Variability Unknown – Range Method
R
LXQ
R
XUQ LU
;
As calculation of Quality indices for Form 2, there is an additional factor c included:
R
cLXQ
R
cXUQ LU
;
A lower specification limit for electrical resistance of a certain electrical
component is 620 ohm. A lot of 100 items is submitted for inspection.
Inspection Level II, normal inspection, with AQL=0.4% is to be used.
From Tables A-2 and C-1 it is seen that a sample of size 10 is
required.Suppose the values of the sample resistance in the order
reading from left to right are as follows:
643, 651, 619, 627, 658 (R1=658-619=39)
670, 673, 641, 638, 650 (R2=673-638=35)
and compliance with the acceptability criterion is to be determined. (using
Form 2)
Q & A
Line Information needed Value Obtained Explanation
1 Sample size, n 10
2 Sum of measurement 6470
3 Sample mean, X-bar 647
4 Average Range, R-bar 37 (39+35)/2
5 Factor c 2.405 Table C-3
6 Spec limit (lower), L 620
7 Quality Index: QL 1.76
8 Est. Lot Percent Ncf below L, pL 2.54% Table C-5
9 Max. Allowable Percent Ncf, M 1.14% Table C-3
10 Acceptability Criteria
a) Compare pL with M
2.45%>1.14%
The lot does not meet the acceptability criterion.
ANSI/ASQ Z1.9
Variability Unknown(Range Method)
Variability Unknown(Standard Deviation Method)
Variability known
Single Specification Double Specification
Form 1 Form 2 Form 2
LSL orUSL
•LSL andUSL•1AQL or2AQL
Quality Index k Quality Index p0 M
k = Acceptability Constant; M = Maximum Allowable Percent Nonconformity
Basic approach is similar to those of Standard Deviation Method.
Master Tables used are Tables D-1 to D-5, instead of B-1 to B-5.
Quality indices for Form 1 are calculated as follows:
Variability known
LX
QXU
Q LU
;
As calculation of Quality indices for Form 2, there is an additional factor v included (Table D-3 and D-4):
vLXQ
vXUQ LU
;
The specified minimum yield point for certain steel castings is 58,000 psi.
A lot of 500 items is submitted for inspection. Inspection Level II, normal
inspection, with AQL=1.5% is to be used. The variability is known to be
300 psi. From Tables A-2 and D-1 it is seen that a sample size of 10 is
required. Suppose the yield points of the sample specimens are:
62,500; 60,500; 68,000; 59,000; 65,500
62,000; 61,000; 69,000; 58,000; 64,500
and compliance with the acceptability criterion is to be determined.
Q & A
Line Information needed Value Obtained Explanation
1 Sample size, n 10
2 Known Varibility: 3,000
2 Sum of measurement 630,000
3 Sample mean, X-bar 63,000
4 Spec limit (lower), L 58,000
5 Quality Index: QL 1.67
6 k 1.70 Table D-1
7 Acceptability Criteria
a) Compare QL with k
1.67%<1.70%
The lot does not meet the acceptability criterion.
Summary - ANSI/ASQ Z1.9
Variability Unknown(Range Method)
Variability Unknown(Standard Deviation Method)
Variability known
Single Specification Double Specification
Form 1 Form 2 Form 2
LSL orUSL
•LSL andUSL•1AQL or2AQL
Quality Index k Quality Index p0 M
k = Acceptability Constant; M = Maximum Allowable Percent Nonconformity
QL or QU kp, pU, or pL M or k
Accept